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	[No title]
		Yet, despite this, the earliest evidence of the golden section is found in an Etruscan dodecahedron from before 500 BC, and similar dodecahedrons have been found elsewhere in the world.
		The first mathematical occurrence of the golden section and it's associated figures is found in the school of thinkers founded by Pythagoras.
		The golden section then sank away into the background as an unimportant curiosity until the work of Leonard Fibonacci of Pisa.
	www.perseus.tufts.edu /GreekScience/Students/Tim/Golden.html (924 words)

	Golden ratio - Wikipedia, the free encyclopedia
		Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e.
		The golden ratio was first studied by ancient mathematicians due to its frequent appearance in geometry and may have even been understood and used as far back in history as the Egyptians.
		Euclid spoke of the "golden mean" this way, "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser".
	en.wikipedia.org /wiki/Golden_ratio (1525 words)

	AllRefer.com - Golden Section (Mathematics) - Encyclopedia
		Golden Section, in mathematics, division of a line segment into two segments such that the ratio of the original segment to the larger division is equal to the ratio of the larger division to the smaller division.
		The Golden Rectangle, whose length and width are the segments of a line divided according to the Golden Section, occupies an important position in painting, sculpture, and architecture, because its proportions have long been considered the most attractive to the eye.
		The numerical ratio of the greater segment of the line to the shorter segment as determined by the Golden Section is symbolized by the Greek letter phi and has the approximate value 1.618.
	reference.allrefer.com /encyclopedia/G/GoldenSe.html (337 words)

	Proportions: Golden Section or Golden Mean, Modulor, Square Root of Two, Theorie and Construction (Site not responding. Last check: 2007-10-26)
		The Golden Section or Golden Mean is derived with simple geometric constructions, its ratio expressed in numbers is, however, irrational ((Square root of five) - 1) : 2 =.618034 : 1 (= 1 : 1.618034).
		The beauty of the golden section may be indicated by the fact that a golden section rectangle subdivides into a square and another, smaller golden section rectangle.
		Without doubt: No. Because of the non-linear nature of the golden section, as clearly demonstrated in the Modulor derivations, it is possible to find some base length and some subdivisions close enough to the ratio of the golden section in anything that may be perceived as beautiful.
	home.att.net /~vmueller/prop/theo.html (1076 words)

	Art
		As the Golden Section is found in the design and beauty of nature, it can also be used to achieve beauty and balance in the design of art.
		Golden sections appear in the stairs and the ring of the trumpet carried by the fourth woman from the top.
		He positioned the two disciples at Christ's side at the golden sections of the width of the composition.
	goldennumber.net /art.htm (588 words)

	The History of the Golden Section (Site not responding. Last check: 2007-10-26)
		The Golden Section, also called the Golden Ratio, the Golden Mean, the Divine Proportion, or the Fibonacci Ratio, is simultaneously one of the most deep and one of the most overdone topics in historical mathematics.
		In the guise of the Fibonacci Sequence, the Golden Section is far-reaching and pervasive in the fields of math, art, architecture, biology, and music.
		The study of the mathematical ratios in plants, and especially of the Golden Section, is called phyllotaxis, and originated in 1754 with a French botanist, C. Bonnet.
	splorg.org:8080 /~b/works/golden (4031 words)

	The Golden Section - Wikipedia, the free encyclopedia
		The Golden Section (1983) is an LP album by English rock-music figure John Foxx.
		His first work with a producer since 1978's Systems Of Romance with Conny Plank, 'The Golden Section was produced by Zeus B Held, well known in the Krautrock scene of the 1970s.
		In addition to echoing his past, The Golden Section also incorporates new elements, like falsetto vocals and lyrical images that seem more LSD-inspired than ever before.
	en.wikipedia.org /wiki/The_Golden_Section (307 words)

	Golden Section
		The Golden Mean is a ratio that is present in the growth patterns of many things--the spiral formed by a shell or the curve of a fern, for example.
		The Golden Mean or Golden Section was derived by the ancient Greeks.
		The Golden Triangle is an isoceles triangle with two angles of 72 degrees and one of 36 degrees.
	www.quantumbalancing.com /golden_section.htm (1529 words)

	Divine proportion the easy way - golden section Photoshop plugin
		Basically it is the division of a line in two sections, where the ratio between the smallest section and the largest section is identical to the ratio between the largest section and the entire length of the line.
		The ratio between the length of your nose and the distance from the bottom of the chin to the bottom of the nose is the golden ratio.
		Here the golden triangles balance the diagonal and the golden spiral enforces the motion upwards right by placing focus on the dramatic point: the index finger pushing up the chin while at the same time taking the motion from the hand entering at the lower right corner.
	powerretouche.com /Divine_proportion_tutorial.htm (1298 words)

	III. The Golden Section
		This implies that the apothema and the semiside are in relation of golden section, that is, that semiside is to the apothema as the apothema is to the sum of the two.
		The conclusion that the Egyptians of the Old Kingdom were acquainted with the golden section is so startling in relation to the current assumptions about the level of Egyptian mathematics, that it could not be accepted merely on the basis of Herodotos’ statement and the actual dimensions of the Great Pyramid.
		The connection between the golden section and the theory of five solids, was emphasized not only by Kepler, but also by Luca Pacioli who in appendix on his treatise on the golden section wrote a treatise on the five bodies inscribed in the sphere.
	www.metrum.org /key/pyramids/third.htm (7550 words)

	American Scientist Online - Did Mozart Use the Golden Section? (Site not responding. Last check: 2007-10-26)
		If Mozart used the golden section, then the other ratio from the golden--section equality--in this case, the ratio of the number of measures in an exposition to those in the recapitulation and development--should also equal the golden ratio.
		The first analysis suggests that Mozart probably did use the golden section, but the variability in the ratios from the second analysis suggests that he did not use the golden section.
		Mozart may have known of the golden section and used it." Nevertheless, Putz thinks that the considerable variation in the data "suggests otherwise." In any case, Mozart did create divine divisions in his piano sonatas-making the interplay of sections shine like sunlight.
	www.americanscientist.org /template/AssetDetail/assetid/24551 (1001 words)

	Golden Section
		The Golden Section is a ratio based on a phi
		The Golden Section is also known as the Golden Mean, Golden Ratio and Divine Proportion.
		It also appears in the physical proportions of the human body, movements in the stock market and many other aspects of life and the universe.
	goldennumber.net /goldsect.htm (224 words)

	Read This: The Golden Section
		As most readers of MAA Online know, the golden section is that which results when a line segment is divided into two pieces such that the ratio of the whole to the larger is the same as the ratio of the larger to the smaller.
		The earliest use of "golden section" was in a German book published in 1835, a consequence, as one commentator put it, of "German romanticism".
		The Golden Section, by Hans Walser, translated by Peter Hilton with the assistance of Jean Pedersen.
	www.maa.org /reviews/golden.html (1751 words)

	A MUSEUM DEDICATED TO THE CONCEPT OF HARMONY AND THE GOLDEN SECTION
		Although the material of every exhibition is well known separately, the collection of facts concerning the golden section confirms the outstanding role it plays in the history of culture.
		The golden section arises from the division of the line-segment
		golden ratio and the division of the line-segment in the ratio of (1) is called the
	www.fenkefeng.org /essaysm18004.html (2811 words)

	The Golden Section (Site not responding. Last check: 2007-10-26)
		The Golden Section has been referred to as the Divine Proportion, the Golden Rectangle, or the Fibonacci Sequence (after Leonardo Fibonacci of Pisa who pioneered some of the early mathematical phenomena and its connection with nature).
		The Golden Section is a simple tool that may be used to enhance the meaning and beauty of an architectural work.
		Most of all, though, the Golden Section reminds us that proportions are important to all works of art, and that we must always learn from the masters of the past.
	www.ewersarchitecture.com /golden_section.htm (484 words)

	Design and Proportion - The 'Golden Mean'
		The 'Golden Mean' is merely a mathematical ratio usually discerned by the painter as the ratio of the larger side of a rectangle as it relates to the shorter.
		In classical architecture it was thought this particular ratio was the most pleasing to the eye and its extrapolation into a spiral could be found replicated in nature in such diverse things as pine cones and sea shells or the curve of a fern.
		To find out how I designed this particular painting format, using the golden mean, you will need to go to the advanced golden mean section of these lessons as the particular details may cause the odd frown or need for some to visit our site refreshment area (open 24 hours).
	www.geocities.com /~jlhagan/lessons/design2.htm (423 words)

	Golden Ratio
		The study of the Golden Ratio allows teachers and learners to explore a well-known problem through four important representations as well as to incorporate technology in a manner that illuminates different aspects of the problem.
		Interest in the Golden Ratio problem may be generated by consulting one of the many articles and books on the subject.
		golden section, when the longer of the two segments formed is the mean proportional between the shorter segment and the whole line.
	www.ite.sc.edu /dickey/golden/golden.html (1253 words)

	Golden Section in Architectural History by Marcus Frings for the Nexus Network Journal vol.4 no.1 (Winter 2002) (Site not responding. Last check: 2007-10-26)
		Marcus Frings traces Golden Section from the mathematical and rather theoretical character of Pacioli's concept, examines Alberti, Serlio, Palladio and other architectural treatises, to arrive to Adolf Zeising in the nineteenth century and to theorist Matila Ghyka and the practitioners Ernst Neufert and Le Corbusier in the twentieth.
		Nor is the Golden Section mentioned in the studies of Vitruvius in the Cinquecento, e.g., Guillaume Philandrier [1544] or Daniele Barbaro [1556].
		None of the lines show the Golden Section; on the contrary they are simply remains of the irregular grids of the former version.
	www.nexusjournal.com /Frings.html (6504 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		The "Golden Proportion" is the one where the ratio of the shorter segment to the longer segment (x : 1-x) is equal to the ratio of the longer segment to the whole segment (1-x : 1).
		The rectangle is a basic form used in art, and the Greeks believed that the most "beautiful" rectangle must therefore utilize the ratio of length to width found in that proportion.
		Draw a diagonal within the "Golden Rectangle." Note that the interior is divided into 2 congruent triangles, each with congruent corresponding angles.
	www.michaelminnerphoto.com /golden.htm (553 words)

	Golden Section in Greek's Art
		The idea of harmony based on the "golden section" became one of the fruitful ideas of the Greek art.
		The angle between theatron and scene divides a circumference of the basis of an amphitheater in ratio: 137°,5 : 222°,5 = 0.618 (the golden proportion).
		The theory of measurement of harmony by a principle of division of the whole in middle and extreme ratio (the "golden section") developed by antique mathematicians became as that foundation, that launch pad, on which concepts of harmony in science and art of European culture subsequently were constructed.
	www.goldenmuseum.com /0305GreekArt_engl.html (1174 words)

	Fibonacci (Site not responding. Last check: 2007-10-26)
		The influence and interconnections between Muslim art the Golden Proportion, the polygons, the Vedic Hindu Square and the Cabbala number system, are beautifully illustrated in the "Language of Pattern" by Alburn, Smith, Steel and Walker published by Thames and Hudson.
		This angle is non other than the Golden Proportion related to the perimeter of a circle as in the adjacent figure.
		The other example of the Golden Proportion is concerned with the number of leaves between one leaf and the next one directly overhead and the number of rotations before this position is reached..
	www.goldenmeangauge.co.uk /fibonacci.htm (1388 words)

	Golden section - Pentagon and Pentagram - Dodecahedron
		Golden section - Pentagon and Pentagram - Dodecahedron
		The length of the diagonals of the pentagons equals the length of the sides of the cube.
		It follows that the length of the edges of the dodecahedron is the greatest of the two parts in which the edge of the cube is divided by the golden section!
	cage.rug.ac.be /~hs/polyhedra/dodeca.html (741 words)

	Links (Site not responding. Last check: 2007-10-26)
		The Golden Mean and a geometric expansion of a line in the proportions of the Golden Section, presented by T. Smith.
		Archaeological speculations on the Mayan Calendar and the Golden Section.
		The ratio of the Golden Section in urban growth documented by the University of Michigan Institute of Mathematical Geography.
	home.att.net /~vmueller/links/links.html (577 words)

	Math Forum: Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence
		A Golden Rectangle is a rectangle in which the ratio of the length to the width is the Golden Ratio.
		The Golden Ratio is the ratio of BC to AB.
		If you have a Golden Rectangle and you cut a square off it so that what remains is a rectangle, that remaining rectangle will also be a Golden Rectangle.
	mathforum.org /dr.math/faq/faq.golden.ratio.html (482 words)

	Spirals and the Golden Section by John Sharp for the Nexus Network Journal vol.4 no.1 (Winter 2002) (Site not responding. Last check: 2007-10-26)
		John Sharp examines spirals and the golden section for the Nexus Network Journal, Winter 2002.
		Since this special issue of the NNJ is concerned with the Golden Section, I am not describing its properties unless appropriate.
		Spirals from the Golden rectangle, triangles and the pentagon by approximation
	www.nexusjournal.com /Sharp_v4n1-intro.html (266 words)

	The Golden Section and the Golden Rectangle
		The golden section was found by the Pythagoreans who used the Pentagram formed, by the diagonals of a regular Pentagon, as a symbol of their school.
		The Pentagon, and the Pythagoras Pentagram Symbol, The symbol of the Golden Section Phi (derived from Pheidias), Plato's Dodecahedron, Archimedes Pi, the Vitruvius Man...
		The golden rectangle is probably the most aesthetic rectangle at least as some tests have shown the result was close to the golden rectangle among different rectangles.
	www.mlahanas.de /Greeks/GoldenSection.htm (1505 words)

	The Golden Mean (Site not responding. Last check: 2007-10-26)
		The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos.
		When you swing the long side of a Golden Rectangle around to create a new rectangle, the line it forms with the short side is made up of two sections having lengths of phi and one, respectively.
		This entire 'Golden Mean' site is dedicated to Liane Hansen of National Public Radio, whose honesty about her unfamiliarity with the 'Golden Section' during a 1995 interview inspired the original version of this page, and to Ken Beattie, who first got me thinking seriously about the signifcance of phi.
	www.vashti.net /mceinc/golden.htm (831 words)

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